cutting face at texture wrap
You could: 


Make a flat plane, and create UVs. 
It should automatically adjust to the texture you have.
Duplicate it, and then move it to each side, so that you create that checker board pattern you want.
Then, you could simply create a cylinder and use it in a boolean operation to cut the flat planes into the circular shape you want. 
You'll need to do some extra cleanup, but it should work.